Journal of Computational Neuroscience

, Volume 45, Issue 3, pp 173–191 | Cite as

Linear-nonlinear-time-warp-poisson models of neural activity

  • Patrick N. LawlorEmail author
  • Matthew G. Perich
  • Lee E. Miller
  • Konrad P. Kording


Prominent models of spike trains assume only one source of variability – stochastic (Poisson) spiking – when stimuli and behavior are fixed. However, spike trains may also reflect variability due to internal processes such as planning. For example, we can plan a movement at one point in time and execute it at some arbitrary later time. Neurons involved in planning may thus share an underlying time course that is not precisely locked to the actual movement. Here we combine the standard Linear-Nonlinear-Poisson (LNP) model with Dynamic Time Warping (DTW) to account for shared temporal variability. When applied to recordings from macaque premotor cortex, we find that time warping considerably improves predictions of neural activity. We suggest that such temporal variability is a widespread phenomenon in the brain which should be modeled.


Modeling Spike trains Poisson process Generalized linear model Reaching movements 


Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

Supplementary material

10827_2018_696_MOESM1_ESM.pdf (1.2 mb)
(PDF 1.20 MB)


  1. Aldworth, Z.N., Miller, J.P., Gedeon, T., Cummins, G.I., Dimitrov, A.G. (2005). Dejittered spike-conditioned stimulus waveforms yield improved estimates of neuronal feature selectivity and spike-timing precision of sensory interneurons. The Journal of Neuroscience, 25(22), 5323–5332. Scholar
  2. Aldworth, Z.N., Dimitrov, A.G., Cummins, G.I., Gedeon, T., Miller, J.P. (2011). Temporal encoding in a nervous system. PLoS Computational Biology, 7(5), e1002041–e1002041. Scholar
  3. Berndt, D., & Clifford, J. (1994). Using dynamic time warping to find patterns in time series. Workshop on Knowledge Knowledge Discovery in Databases, 398, 359–370.Google Scholar
  4. Buesing, L., Macke, J.H., Sahani, M. (2012). Learning stable, regularised latent models of neural population dynamics. Network: Computation in Neural Systems, 23(1-2), 24–47. Scholar
  5. Carandini, M. (2004). Amplification of trial-to-trial response variability by neurons in visual cortex. PLoS Biology, 2(9), e264–e264. Scholar
  6. Chase, S.M., Schwartz, A.B., Kass, R.E. (2010). Latent inputs improve estimates of neural encoding in motor cortex. The Journal of Neuoscience, 30(41), 13,873–13,882. Scholar
  7. Churchland, M.M., & Shenoy, K.V. (2007). Temporal complexity and heterogeneity of single-neuron activity in premotor and motor cortex. Journal of Neurophysiology, 97(6), 4235–4257. Scholar
  8. Cisek, P., & Kalaska, J.F. (2004). Neural correlates of mental rehearsal in dorsal premotor cortex. Nature, 431(7011), 993–996. Scholar
  9. Cisek, P., & Kalaska, J.F. (2005). Neural correlates of reaching decisions in dorsal premotor cortex: specification of multiple direction choices and final selection of action. Neuron, 45(5), 801–814. Scholar
  10. Cohen, M.R., & Maunsell, J.H.R. (2009). Attention improves performance primarily by reducing interneuronal correlations. Nature Neuroscience, 12(12), 1594–1600. Scholar
  11. Crammond, D.J., & Kalaska, J.F. (2000). Prior information in motor and premotor cortex: activity during the delay period and effect on pre-movement activity. Journal of Neurophysiology, 84(2), 986–1005.PubMedCrossRefGoogle Scholar
  12. de Ruyter van Steveninck, RR, Lewen, G.D., Strong, S.P., Köberle, R., Bialek, W. (1997). Reproducibility and variability in neural spike trains. Science, 275(5307), 1805–1808.
  13. Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 38, 1–38.CrossRefGoogle Scholar
  14. Fernandes, H.L., Stevenson, I.H., Phillips, A.N., Segraves, M.A., Kording, K.P. (2013). Saliency and saccade encoding in the frontal eye field during natural scene search. Cerebral cortex (New York NY), 1991, 1–14. Scholar
  15. Gold, J.I., & Shadlen, M.N. (2007). The neural basis of decision making. Annual Review of Neuroscience, 30, 535–574. Scholar
  16. Gollisch, T. (2006). Estimating receptive fields in the presence of spike-time jitter. Network (Bristol England), 17(2), 103–129. Scholar
  17. Goris, R.L.T, Movshon, J.A., Simoncelli, E.P. (2014). Partitioning neuronal variability. Nature Neuroscience (April).
  18. Guo, Z.V., Inagaki, H.K., Daie, K., Druckmann, S., Gerfen, C.R., Svoboda, K. (2017). Maintenance of persistent activity in a frontal thalamocortical loop. Nature, 545(7653), 181–186. Scholar
  19. Haith, A.M., Pakpoor, J., Krakauer, J.W. (2016). Independence of movement preparation and movement initiation. Journal of Neuroscience, 36(10), 3007–3015. Scholar
  20. Hubel, D.H., & Wiesel, T.N. (1962). Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160, 106–154.PubMedPubMedCentralCrossRefGoogle Scholar
  21. Kisley, M.A., & Gerstein, G.L. (1999). Trial-to-trial variability and state-dependent modulation of auditory-evoked responses in cortex. Journal of Neuroscience, 19(23), 10,451–10,460.CrossRefGoogle Scholar
  22. Kollmorgen, S., & Hahnloser, R.H.R. (2014). Dynamic alignment models for neural coding. PLoS Computational Biology, 10(3), e1003508–e1003508. Scholar
  23. Lakshmanan, K.C., Sadtler, P.T., Tyler-Kabara, E.C., Batista, A.P., Yu, B.M. (2015). Extracting low-dimensional latent structure from time series in the presence of delays. Neural Computation, 27(9), 1825–1856. Scholar
  24. Latimer, K.W., Yates, J.L., Meister, M.L.R., Huk, A.C., Pillow, J.W. (2015). Single-trial spike trains in parietal cortex reveal discrete steps during decision-making. Science, 349(6244), 184–187. Scholar
  25. Lawhern, V., Wu, W., Hatsopoulos, N., Paninski, L. (2010). Population decoding of motor cortical activity using a generalized linear model with hidden states. Journal of Neuroscience Methods, 189(2), 267–280. Scholar
  26. Lin, I.C., Okun, M., Carandini, M., Harris, K. D. (2015). The nature of shared cortical variability. Neuron, 87(3), 1–13. Scholar
  27. Mitchell, J.F., Sundberg, K.A., Reynolds, J.H. (2009). Spatial attention decorrelates intrinsic activity fluctuations in macaque area V4. Neuron, 63(6), 879–888. Scholar
  28. Nelder, J.A., & Baker, R.J. (1972). Generalized linear models. Encyclopedia of Statistical Sciences.Google Scholar
  29. Nordstrom, M., Fuglevand, A., Enoka, R. (1992). Estimating the strength of common input to human motoneurons from the cross-correlogram. The Journal of Physiology, 453, 547–574.PubMedPubMedCentralCrossRefGoogle Scholar
  30. Okun, M., Steinmetz, N.A., Cossell, L., Iacaruso, M.F., Ko, H., Barthó, P., Moore, T., Hofer, S.B., Mrsic-Flogel, T.D., Carandini, M., Harris, K.D. (2015). Diverse coupling of neurons to populations in sensory cortex. Nature.
  31. Perez, O., Kass, R.E., Merchant, H. (2013). Trial time warping to discriminate stimulus-related from movement-related neural activity. Journal of Neuroscience Methods, 212(2), 203–210. Scholar
  32. Pfau, D., Pnevmatikakis, E.A., Paninski, L. (2013). Robust learning of low-dimensional dynamics from large neural ensembles. In Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (Eds.) Advances in neural information processing systems (Vol. 26, pp. 2391–2399). Red Hook: Curran Associates, Inc.Google Scholar
  33. Pillow, J.W., Shlens, J., Paninski, L., Sher, A., Litke, A.M., Chichilnisky, E.J., Simoncelli, E.P. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207), 995–999. Scholar
  34. Rabinowitz, N.C., Goris, R.L., Cohen, M., Simoncelli, E. (2015). Attention stabilizes the shared gain of V4 populations. eLife, 4, e08998–e08998. Scholar
  35. Ramkumar, P., Lawlor, P.N., Glaser, J.I., Wood, D.K., Phillips, A.N., Segraves, M.A., Kording, K.P. (2016). Feature-based attention and spatial selection in frontal eye fields during natural scene search. Journal of Neurophysiology, 116 (3), 1328–1343. Scholar
  36. Reich, D.S., Victor, J.D., Knight, B.W., Ozaki, T., Kaplan, E. (1997). Response variability and timing precision of neuronal spike trains in vivo. Journal of Neurophysiology, 77(5), 2836–2841.PubMedCrossRefGoogle Scholar
  37. Sakoe, H., & Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. ASSP-26, 26(1), 43–49. Scholar
  38. Shenoy, K.V., Sahani, M., Churchland, M.M. (2013). Cortical control of arm movements: a dynamical systems perspective. Annual Review of Neuroscience, 36(1), 337–359. Scholar
  39. Siegel, M., Buschman, T.J., Miller, E.K. (2015). Cortical information flow during flexible sensorimotor decisions. Science, 348(6241), 1352–1355. Scholar
  40. Stevenson, I.H., Rebesco, J.M., Miller, L.E., Körding, K.P. (2008). Inferring functional connections between neurons. Current Opinion in Neurobiology, 18(6), 582–588. Scholar
  41. Stevenson, I.H., London, B.M., Oby, E.R., Sachs, N.A., Reimer, J., Englitz, B., David, S.V., Shamma, S.A., Blanche, T.J., Mizuseki, K., Zandvakili, A., Hatsopoulos, N.G., Miller, L.E., Kording, K.P. (2012). Functional connectivity and tuning curves in populations of simultaneously recorded neurons . PLoS Computational Biology, 8(11), e1002775–e1002775. Scholar
  42. Ventura, V., Cai, C., Kass, R.E. (2005). Trial-to-trial variability and its effect on time-varying dependency between two neurons. Journal of Neurophysiology, 94(4), 2928–2939.PubMedCrossRefGoogle Scholar
  43. Victor, J.D. (2005). Spike train metrics. Current Opinion in Neurobiology, 15(5), 585–592. Scholar
  44. Victor, J.D., & Purpura, K.P. (1996). Nature and precision of temporal coding in visual cortex: a metric-space analysis. Journal of Neurophysiology, 76(2), 1310–1326.PubMedCrossRefGoogle Scholar
  45. Vidne, M., Ahmadian, Y., Shlens, J., Pillow, J.W., Kulkarni, J., Litke, A.M., Chichilnisky, E.J., Simoncelli, E., Paninski, L. (2012). Modeling the impact of common noise inputs on the network activity of retinal ganglion cells. Journal of Computational Neuroscience, 33(1), 97–121. Scholar
  46. Weinrich, M., Wise, S.P., Mauritz, K.H. (1984). A neurophysiological study of the premotor cortex in the rhesus monkey. Brain: A Journal of Neurology, 2, 385–414. Scholar
  47. Wurtz, R.H. (1969a). Comparison of effects of eye movements and stimulus movements on striate cortex neurons of the monkey. Journal of Neurophysiology, 32(98), 994–994.Google Scholar
  48. Wurtz, R.H. (1969b). Visual receptive fields of striate cortex neurons in awake monkeys. Journal of Neurophysiology, 32(5), 727– 742.Google Scholar
  49. Yu, B.M., Cunningham, J.P., Santhanam, G., Ryu, S.I., Shenoy, K.V., Sahani, M. (2009). Gaussian-process factor analysis for low-dimensional single-trial analysis of neural population activity. Journal of Neurophysiology, 102(1), 614–635. Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Child NeurologyChildren’s Hospital of PhiladelphiaPhiladelphiaUSA
  2. 2.University of GenevaGenevaSwitzerland
  3. 3.Department of PhysiologyNorthwestern UniversityChicagoUSA
  4. 4.Departments of Bioengineering and NeuroscienceUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations